sin²θ /Cos²θ + cosθ= secθ

Solution

We can start by manipulating the left-hand side of the equation using trigonometric identities.

sin²θ / (cos²θ + cosθ)

= sin²θ / cosθ(cosθ + 1)

= sinθ/cosθ * sinθ/cosθ / (1 + cosθ)

= tan²θ / (1 + cosθ)

Now, we can use the definition of secant:

secθ = 1/cosθ

Multiplying the numerator and denominator of the right-hand side by cos²θ, we get:

tan²θ / (1 + cosθ) * cos²θ/cos²θ

= sin²θ/cos²θ * cos²θ/(1 + cosθ)

= (sinθ/cosθ)² / (1/cosθ)(1 + cosθ)

= (tanθ)² / (1 + cosθ)

Now we can substitute this expression for the left-hand side of the original equation:

sin²θ / (cos²θ + cosθ) = (tanθ)² / (1 + cosθ)

= sec²θ – 1 / (1 + cosθ) (using the identity tan²θ + 1 = sec²θ)

= (1/cos²θ) – 1 / (1 + cosθ)

= (1 – cos²θ) / (cos²θ * (1 + cosθ))

= sin²θ / (cos²θ + cosθ)

Therefore, sin²θ / (cos²θ + cosθ) = secθ.